Chapter 3: Trusses – Finite Element Analysis and Applications

Trusses

Introduction

This chapter introduces the concept of trusses, their structural significance, and the finite element method (FEM) for analyzing truss systems. The content is essential for understanding how to model, analyze, and verify truss structures in engineering and physics contexts.

Definition of a Truss

What is a Truss?

• Truss: An engineering structure composed of straight members connected at their ends, typically by bolts, rivets, pins, or welding.

• Trusses are widely used in practical engineering applications such as:

  • Power transmission towers

  • Bridges

  • Roofs of buildings

Plane Truss/Simple Truss

• A plane truss is a truss whose members and applied forces all lie in a single plane.

• Members are generally considered two-force members, meaning internal forces act in equal and opposite directions along the member's axis.

• Members experience either tension (pulling apart) or compression (pushing together).

Example: A simple triangular truss supporting a load at the apex, with two members in compression and one in tension.

Assumptions for Plane Truss Members

• Members are connected by frictionless pins.

• Members are subjected only to axial forces (tension or compression).

• Members cannot develop moments at their ends.

• Loads act only at the joints (nodes).

• Each member has a uniform cross-section.

• The weight of the truss is negligible compared to the applied load.

Statically Determinate and Indeterminate Trusses

Statically Determinate Truss Problems

• Can be solved using only the equations of static equilibrium.

• Analyzed by the method of joints or method of sections.

• Do not provide information about joint deflections, as members are treated as rigid bodies.

Statically Indeterminate Truss Problems

• Cannot be solved by static equilibrium equations alone due to extra unknowns.

• Require additional compatibility equations, often solved using the finite element method (FEM).

• FEM allows for the analysis of joint displacements and internal forces in statically indeterminate trusses.

Comparison Table: Statically Determinate vs. Indeterminate Trusses

Type	Unknown Reactions	Equilibrium Equations	Solvability
Statically Determinate	3	3	Solvable by statics alone
Statically Indeterminate	4 or more	3	Requires additional equations (e.g., FEM)

Finite Element Formulation of Truss Analysis

Overview

The finite element method (FEM) is a numerical technique for finding approximate solutions to boundary value problems. In truss analysis, FEM is used to determine joint displacements and internal forces.

Stiffness of a Truss Member

• The average stress in a member: $\sigma = \frac{P}{A}$

• The average normal strain: $\epsilon = \frac{\Delta L}{L}$

• Hooke's Law for axial deformation: $\sigma = E \epsilon$

• Axial deformation: $\Delta L = \frac{P L}{A E}$

• Equivalent spring constant: $k_{eq} = \frac{A E}{L}$

• Force-displacement relation: $F = k_{eq} x$

Coordinate Systems in Truss Analysis

• Global coordinate system (XY): Used to define the position of each node and the orientation of each element.

• Local coordinate system (xy): Used to describe the behavior of individual members.

• Transformation between local and global coordinates is necessary for assembling the global stiffness matrix.

Transformation of Displacements and Forces

• Global displacements $(U_{iX}, U_{iY}, U_{jX}, U_{jY})$ at nodes $i$ and $j$ are related to local displacements $(u_{ix}, u_{iy}, u_{jx}, u_{jy})$ by:

$\begin{align*} U_{iX} &= u_{ix} \cos \theta - u_{iy} \sin \theta \\ U_{iY} &= u_{ix} \sin \theta + u_{iy} \cos \theta \\ U_{jX} &= u_{jx} \cos \theta - u_{jy} \sin \theta \\ U_{jY} &= u_{jx} \sin \theta + u_{jy} \cos \theta \end{align*}$

Or in matrix form:

$\begin{pmatrix} U_{iX} \\ U_{iY} \\ U_{jX} \\ U_{jY} \end{pmatrix} = \begin{pmatrix} \cos \theta & -\sin \theta & 0 & 0 \\ \sin \theta & \cos \theta & 0 & 0 \\ 0 & 0 & \cos \theta & -\sin \theta \\ 0 & 0 & \sin \theta & \cos \theta \end{pmatrix} \begin{pmatrix} u_{ix} \\ u_{iy} \\ u_{jx} \\ u_{jy} \end{pmatrix}$

Matrix Relationships in FEM

• Displacement transformation: $\{U\} = [T]\{u\}$

• Force transformation: $\{F\} = [T]\{f\}$

• Inverse transformations: $\{u\} = [T]^{-1}\{U\}$, $\{f\} = [T]^{-1}\{F\}$

• Element force-displacement relation: $\{f\} = [k]\{u\}$

• Global force-displacement relation: $\{F\} = [K]\{U\}$, where $[K] = [T][k][T]^{-1}$

Summary of Key Concepts

• Trusses are essential structural elements analyzed using static equilibrium and, for more complex cases, the finite element method.

• Understanding the difference between statically determinate and indeterminate trusses is crucial for selecting the appropriate analysis method.

• The finite element method provides a systematic approach to determine displacements and internal forces in truss structures.

• Transformation between local and global coordinate systems is necessary for assembling and solving the global stiffness matrix.

Additional info: The notes also cover practical steps for solving truss problems using FEM software (e.g., ANSYS), including preprocessing (model setup), solution (solving equations), and postprocessing (interpreting results and verification). These steps are standard in computational structural analysis.